Continuity equation — derivation (conservation of mass), ρAv = const: Question bank

Before we start, one reminder of the two forms so every symbol below is earned:

Spot the error

SE1. "Water speeds up in a narrow pipe, so its mass flow rate goes up there too."

Wrong — mass flow rate

ρ A v

is the conserved quantity; the speed rises exactly to keep

ρ A v

constant, it does not increase.

SE2. "Since

Q = A v

and

A

drops,

Q

drops in the narrow section."

Wrong — for incompressible flow

Q

is constant; as

A

drops,

v

rises so their product

Q

stays fixed.

SE3. "

A \propto r

, so halving the radius halves the area."

Wrong — a circular cross-section has

A = π r^{2}

, so area

\propto r^{2}

; halving

r

makes

A

one-quarter.

SE4. "The bucket example uses

v^{2}

, so it's a Bernoulli problem."

Wrong — filling time uses only

Q = A v

(a continuity/volume-flow idea); no pressure or energy balance appears.

SE5. "Mass in doesn't have to equal mass out — fluid can pile up in the middle."

Wrong in steady incompressible flow — the region is already full and

ρ

can't rise, so accumulation is impossible; in = out.

SE6. "In example 3 the area was the same, so nothing should change speed."

Wrong — density doubled, and

ρ A v

constant with fixed

A

forces

v

to halve; density can change the speed even at constant area.

SE7. "Continuity needs the fluid to be frictionless and ideal."

Wrong — continuity is pure mass bookkeeping; it holds for viscous, real fluids too. Only Bernoulli needs the ideal-flow assumptions.

Why questions

WHY1. Why does squeezing a hose end make the water shoot out faster?

The same volume per second must pass a smaller opening, so by

A v = Q = const

the speed

v

must rise as

A

falls.

WHY2. Why do we cancel

Δ t

in the derivation instead of keeping it?

Both sides carried the same

Δ t

(mass in

= ρ_{1} A_{1} v_{1} Δ t

, out

= ρ_{2} A_{2} v_{2} Δ t

); dividing it out gives a relation true for any time interval.

WHY3. Why is

ρ A v

preferred over

A v

as the "master" statement?

Because

ρ A v

(mass flow) is conserved for all fluids;

A v

is just the special case that survives when

ρ

cancels for incompressible flow.

WHY4. Why does continuity alone not tell you the pressure in the narrow section?

Continuity fixes speeds from geometry and mass, but pressure comes from energy balance — that requires Bernoulli's equation.

WHY5. Why can a river run fast and shallow over rocks yet slow and deep in a pool?

The discharge

Q = A v

is roughly constant, so where the cross-section (depth × width) shrinks the water speeds up, and vice-versa.

WHY6. Why is a Venturi meter built with a deliberate narrowing?

The narrow throat forces a known speed-up via

A_{1} v_{1} = A_{2} v_{2}

; measuring the resulting pressure drop then reveals the flow rate.

WHY7. Why must we assume "no mass accumulates" for the simple form to hold?

If mass built up between sections, inflow would exceed outflow and

ρ_{1} A_{1} v_{1} = ρ_{2} A_{2} v_{2}

would fail; steadiness is what makes in = out.

Edge cases

EC1. What does continuity say for a completely blocked pipe (

v = 0

everywhere)?

ρ A v = 0

at every section — a valid, consistent solution; zero in equals zero out, no motion.

EC2. What happens to the exit speed as the outlet area shrinks toward zero (incompressible)?

v_{2} = v_{1} A_{1} / A_{2} \to \infty

; the model predicts unbounded speed, warning that real effects (viscosity, cavitation) must eventually intervene.

EC3. If the fluid is incompressible AND the pipe area is uniform, what does continuity reduce to?

Simply

v_{1} = v_{2}

; a trivial but correct statement — constant speed, nothing to solve.

EC4. Can the mass flow rate

ρ A v

ever differ between two sections of the same steady tube?

No — steadiness plus no leaks makes

ρ A v

identical at every section; that constancy is the continuity equation.

EC5. For a steadily heated gas that expands (drops

ρ

) in a constant-area duct, what must happen to

v

It must rise: with

A

fixed and

ρ

falling,

v = \overset{m}{˙} / (ρ A)

increases to keep mass flow constant — see Incompressible vs compressible flow.

EC6. Two identical hoses feed one pipe of the same total area — does the merged speed equal one hose's speed?

No — the combined mass flow is doubled, so if the merged area equals one hose's area the speed doubles; areas and flows both must be balanced.

EC7. In steady flow, if density and area are both constant but the fluid still accelerates, is continuity violated?

Yes it would be — constant

ρ

and

A

force constant

v

; a genuine acceleration means either

A

ρ

is actually changing somewhere.

Recall One-line summary of the traps

The single sentence that defuses almost every trap here: == $ρ A v$ is what nature keeps fixed; speed changes only to protect it.== Wide⇒slow, narrow⇒fast, and drop $ρ$ only when you're sure the fluid is incompressible.

Connections

Parent: Continuity equation
Conservation of mass — the law behind every answer above.
Incompressible vs compressible flow — decides which form you may use.
Bernoulli's equation — where continuity hands off to energy.
Venturi meter · Volume flow rate and discharge · Streamlines and stream tubes

Question bank — Continuity equation — derivation (conservation of mass), ρAv = const

True or false — justify